#7712024

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,3,4)(7,11)(10,12)(13,14)(15,20)(16,18)(17,19)(21,25)(22,27)(23,24)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,25)(14,20,17,21)(16,23,22,26)(18,24,27,28);(6,9)(10,12)(13,15)(14,21)(16,26)(17,20)(18,24)(19,25)(22,23)(27,28);(13,16)(14,18)(15,23)(17,27)(19,22)(20,24)(21,28)(25,26);(5,6)(7,10)(8,9)(11,12)(13,17,19,14)(15,21,25,20)(16,27,22,18)(23,28,26,24);(5,7,8,11)(6,10,9,12)(13,18,19,27)(14,22,17,16)(15,24,25,28)(20,26,21,23)

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619220);